Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 26. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Quadratics and Their Graphs 231 EXAMPLE SOLUTION QUADRATICS AND THEIR GRAPHS Graphing Parabolas Beginning with the basic parabola y x2 and using the results of Section on shifting stretching shrinking and flipping we can obtain any other parabola in which we are interested. Graph x 2 x - 4 2 - 2. This is the graph of y x2 shifted to the right 4 units shrunk vertically to half the height at each x-value and then shifted downward by 2 units. Use the same order of operations graphically as you would algebraically .8 In particular the vertex of y x2 is shifted right 4 units to 4 0 shrunk vertically no effect because zero can t be shrunk and then shifted down 2 units to 4 -2 . A graph is shown below. EXERCISE Figure Using analogous lines of reasoning we can graph any function of the form x a x - A 2 . In fact by using the algebraic technique of completing the square any quadratic can be expressed in this form. 9 Suppose we are given a quadratic in the form x a x - r x - 5 where a r and 5 are constants. This is also convenient to graph. We can immediately read off the zeros of the function x 0 when x r or x 5. If a 0 then the parabola opens upward and if a 0 then the parabola opens downward. Only parabolas that intersect the x-axis can be expressed in this form. Find the equation of the family of parabolas with vertex 5 0 . Your answer should involve one parameter that is one unspecified constant. Suppose the quadratic in Example was not given in the form x a x - A 2 suppose we were given this same function in the form 1 2 . x x2 - 4x 6. To sketch this by hand there are a couple of options. One which we won t use here is to put this equation in the form x a x - A 2 a process that involves what is called 8 For additional work on order of operations refer to Appendix A Algebra. 9See the derivation of the quadratic formula in the Algebra Appendix to see how this is done. 232 CHAPTER 6 The Quadratics A Profile of a Prominent Family of Functions .
